On universality of blow-up profile for L2 critical nonlinear Schrödinger equation

We consider finite time blow-up solutions to the critical nonlinear Schrödinger equation iut=-Δu-|u|4/Nu with initial condition u0∈H1. Existence of such solutions is known, but the complete blow-up dynamic is not understood so far. For a specific set of initial data, finite time blow-up with a universal sharp upper bound on the blow-up rate has been proved in [22], [23].We establish in this paper the existence of a universal blow-up profile which attracts blow-up solutions in the vicinity of blow-up time. Such a property relies on classification results of a new type for solutions to critical NLS. In particular, a new characterization of soliton solutions is given, and a refined study of dispersive effects of (NLS) in L2 will remove the possibility of self similar blow-up in energy space H1.

[1]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

[2]  Gadi Fibich,et al.  A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation , 1998 .

[3]  Frank Merle,et al.  Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation , 2002 .

[4]  Papanicolaou,et al.  Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. , 1988, Physical review. A, General physics.

[5]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[6]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

[7]  M. Weinstein.  Nonlinear Schrödinger equations and sharp interpolation estimates , 1983 .

[8]  Jean Bourgain,et al.  Construction of blowup solutions for the nonlinear Schr ? odinger equation with critical nonlineari , 1997 .

[9]  Catherine Sulem,et al.  The nonlinear Schrödinger equation , 2012 .

[10]  On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation , 1993 .

[11]  Frank Merle,et al.  A Liouville theorem for the critical generalized Korteweg–de Vries equation , 2000 .

[12]  J. Ginibre,et al.  On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case , 1979 .

[13]  Y. Nakamura,et al.  On nonlinear Schrödinger equations with stark effect , 2002 .

[14]  T. Cazenave,et al.  Some remarks on the nonlinear Schrödinger equation in the subcritical case , 1989 .

[15]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

[16]  Frank Merle,et al.  Existence of blow-up solutions in the energy space for the critical generalized KdV equation , 2001 .

[17]  YeYaojun.  GLOBAL SOLUTIONS OF NONLINEAR SCHRODINGER EQUATIONS , 2005 .

[18]  F. Merle,et al.  Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power , 1993 .

[19]  W. Rother,et al.  Nonlinear scalar field equations , 1992, Differential and Integral Equations.

[20]  Basilis Gidas,et al.  A priori bounds for positive solutions of nonlinear elliptic equations , 1981 .

[21]  Frank Merle,et al.  Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation , 2003 .

[22]  Paul H. Rabinowitz,et al.  On a class of nonlinear Schrödinger equations , 1992 .

[23]  F. Merle,et al.  Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view , 2000 .

[24]  G. Perelman.  On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D , 2000 .

[25]  Pierre Raphaël,et al.  The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation , 2005 .

[26]  G. Perelman.  On the Formation of Singularities in Solutions of the Critical Nonlinear Schrödinger Equation , 2001 .

[27]  W. Zachary,et al.  Nonlinear Semigroups, Partial Differential Equations and Attractors , 1987 .

[28]  Hayato Nawa,et al.  ASYMPTOTIC AND LIMITING PROFILES OF BLOWUP SOLUTIONS OF THE NONLINEAR SCHRODINGER EQUATION WITH CRITICAL POWER , 1999 .

[29]  Frank Merle,et al.  Blow up in finite time and dynamics of blow up solutions for the L^2-critical generalized KdV equation , 2002 .

[30]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .